

A254574


Number of ways to write n = x*(x+1)/2 + y*(3*y+1)/2 + z*(3*z1)/2 with x,y,z nonnegative integers


8



1, 2, 2, 3, 2, 2, 3, 3, 5, 2, 3, 3, 3, 5, 2, 6, 3, 5, 5, 2, 4, 3, 8, 4, 3, 4, 4, 6, 6, 6, 7, 3, 4, 5, 3, 6, 5, 8, 5, 4, 6, 8, 5, 8, 5, 5, 4, 6, 10, 1, 7, 6, 10, 5, 4, 7, 6, 7, 9, 6, 6, 6, 8, 10, 4, 7, 5, 9, 7, 7, 4
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OFFSET

0,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 49.
(ii) Any nonnegative integer not equal to 18 can be written as x*(x+1)/2 + y*(3*y+1) + z*(3*z1) with x,y,z nonnegative integers.
See also the comments of A254573 for a similar conjecture. We have proved that any nonnegative integer can be written as x*(x+1)/2 + y*(3*y+1)/2 + z*(3*z1)/2 with x,y,z integers.
Note that ZhiWei Sun conjectured in 2009 (cf. Conjecture 1.10 of arXiv:0905.0635) that every n = 0,1,... can be expressed as the sum of a triangular number and two pentagonal numbers.


LINKS



EXAMPLE

a(14) = 2 since 14 = 0*1/2 + 1*(3*1+1)/2 + 3*(3*31)/2 = 3*4/2 + 2*(3*2+1)/2 + 1*(3*11)/2.
a(49) = 1 since 49 = 1*2/2 + 4*(3*4+1)/2 + 4*(3*41)/2.


MATHEMATICA

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0; Do[If[TQ[ny(3y+1)/2z(3z1)/2], r=r+1], {y, 0, (Sqrt[24n+1]1)/6}, {z, 0, (Sqrt[24(ny(3y+1)/2)+1]+1)/6}];
Print[n, " ", r]; Continue, {n, 0, 70}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



